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International Journal of Quantum Information

Phase space Heisenberg-limited estimation of the average phase shift in a Mach–Zehnder interferometer

Dario Gatto, Paolo Facchi A, Vincenzo Tamma

Published Feb 1, 2020• 10 citations
Publisher

Abstract

We address the problem of distributed quantum metrology with a single squeezed-vacuum source by using the formalism of quantum mechanics in phase space. In particular, we demonstrate Heisenberg-limited sensitivity in the measurement of the average of two arbitrary phase shifts in the arms of a Mach–Zehnder interferometer. We obtain exact results for the measurement probability at the interferometer output for any value of the phases, which give us insight into the emergence of Heisenberg-limited sensitivity for periodical values of the phases.

DOI
10.1142/s0219749919410193

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We address the problem of distributed quantum metrology with a single squeezed-vacuum source by using the formalism of quantum mechanics in phase space. In particular, we demonstrate Heisenberg-limited sensitivity in the measurement of the average of two arbitrary phase shifts in the arms of a Mach–Zehnder interferometer. We obtain exact results for the measurement probability at the interferometer output for any value of the phases, which give us insight into the emergence of Heisenberg-limited sensitivity for periodical values of the phases.
Abstract The Mach–Zehnder interferometer is a fundamental tool for measuring phase shifts between two light paths, serving as a crucial prototype for achieving high-precision measurements in various scientific and technological applications. In this study, we analyze different models for estimating the relative phase shift in a general two-arm Mach–Zehnder interferometer. We demonstrated that single-parameter estimation models can be reduced from the two-parameter estimation model by imposing appropriate constraints on the parameter space. To make quantum Fisher information of the single-parameter estimation models meaningful, the corresponding constraints must be guaranteed in the experiment implementation. Furthermore, we apply the quantum Fisher information approach to analyze the Mach–Zehnder interferometer with the an input state composed of a displaced squeezed vacuum state and a coherent state, providing insights into the precision limits of such configurations.
We propose a quantum metrological protocol based on a Mach-Zehnder interferometer with a squeezed vacuum input state and an antisqueezing operation at one of its output channels. A simple and intuitive geometrical picture of the state evolution is provided by the marginal Wigner functions of the state at each interferometer output channel. The protocol allows us to detect the values of the sum $\ensuremath{\beta}=\frac{1}{2}({\ensuremath{\varphi}}_{1}+{\ensuremath{\varphi}}_{2})+{\ensuremath{\theta}}_{\mathrm{in}}\ensuremath{-}{\ensuremath{\theta}}_{\mathrm{out}}$, of the relative phase ${\ensuremath{\theta}}_{\mathrm{in}}\ensuremath{-}{\ensuremath{\theta}}_{\mathrm{out}}$ between the two squeezers, and of the average of the phase delays ${\ensuremath{\varphi}}_{1}$ and ${\ensuremath{\varphi}}_{2}$ in the two arms of the interferometer. The detection sensitivity scales at the Heisenberg limit and, remarkably, is robust not only to detector inefficiencies but also to any photon losses occurring before the antisqueezing operation. Interestingly, we demonstrate that in the latter case an increase of sensitivity can even occur by increasing the losses in a suitable range.
Many works have stated that nonlinear interactions can improve phase sensitivity beyond the Heisenberg limit scaling of 1/N1/N1/N with NNN being the mean photon number. This raises some open questions---among them the conclusive sensitivity limits with respect to single-mode inputs. Namely, when one of two inputs is vacuum, is there a shot-noise-style sensitivity bound on a nonlinear Mach-Zehnder interferometer? Within the reach of second-order nonlinear phase shifts, we make an attempt to provide an answer to this question. Based upon phase-averaging approach, this puzzle is partially resolved with careful calculations of the quantum Fisher information regarding three kinds of common inputs: Gaussian states, squeezed number states, and Schrödinger cat states. The results suggest that shot-noise-style sensitivity limit is no longer available, and the ideal candidate is squeezed vacuum.
In 1981, Caves pointed out that the phase sensitivity of a Mach-Zehnder interferometer with single-mode inputs is bounded by the shot-noise limit. The quantum Fisher information analysis shows that this statement holds true for the scenario where two antisymmetric phase shifts occur in two arms, but it is invalid for the scenario where an unknown phase is embedded in one of two arms. In this paper, we focus on the phase sensitivity directed against the latter scenario. The optimal single-mode input is discussed by analyzing common states, including displaced squeezed states, displaced number states, squeezed number states, Schrödinger cat states and completely mixed states. We find that the best choice is a squeezed vacuum state and show the specific measurement scheme which is capable of saturating the corresponding phase sensitivity limit. In addition, we study the effects of several realistic factors-anti-squeezing noise, photon loss and dark counts-on the phase sensitivity. Our results suggest that sub-shot-noise-limited phase sensitivity is attainable with low noise or loss, which paves the way for practical metrology.
Abstract We study the sensitivity of a Mach–Zehnder interferometer that contains in addition to the phase shifter a non-linear element. By including both elements in a cavity or a loop that the light transverses many times, a non-linear kicked version of the interferometer arises. We study its sensitivity as function of the phase shift, the kicking strength, the maximally reached average number of photons, and damping due to photon loss for an initial coherent state. We find that for vanishing damping Heisenberg-limited scaling of the sensitivity arises if squeezing dominates the total photon number. For small to moderate damping rates the non-linear kicks can considerably increase the sensitivity as measured by the quantum Fisher information per unit time.